Problem 78 - Applied Statistics and Probability for Engineers

Question

The rate of return of an asset is the change in price divided by the initial price (denoted as \(r\) ). Suppose that \(\$ 10,000\) is used to purchase shares in three stocks with rates of returns \(X_{1}, X_{2}, X_{3}\). Initially, \(\$ 2500, \$ 3000,\) and \(\$ 4500\) are allocated to each one, respectively. After one year, the distribution of the rate of return for each is normally distributed with the following parameters: \(\mu_{1}=0.12, \sigma_{1}=0.14, \mu_{2}=0.04, \sigma_{2}=0.02, \mu_{3}=0.07, \sigma_{3}=0.08\) (a) Assume that these rates of return are independent. Determine the mean and variance of the rate of return after one year for the entire investment of \(\$ 10,000\). (b) Assume that \(X_{1}\) is independent of \(X_{2}\) and \(X_{3}\) but that the covariance between \(X_{2}\) and \(X_{3}\) is \(-0.005 .\) Repeat part (a). (c) Compare the means and variances obtained in parts (a) and (b) and comment on any benefits from negative covariances between the assets.

Step-by-Step Solution

Answer

Mean rate of return is 0.084 in both parts. Variance decreases from 0.00509 to 0.00464 due to negative covariance benefit.

Step 1: Calculate Weighted Means of Returns

To find the mean rate of return for the entire investment, consider the weighted average with the allocated values. For part (a), using weights of \(2500/10000\), \(3000/10000\), and \(4500/10000\), calculate:\[ \mu = \frac{2500}{10000} \times 0.12 + \frac{3000}{10000} \times 0.04 + \frac{4500}{10000} \times 0.07 \].Evaluating this gives:\[ \mu = 0.084 \].

Step 2: Calculate Variance for Independent Returns

In part (a), the variances add up according to their weights since the assets are independent:\[ \sigma^2 = \left( \frac{2500}{10000} \right)^2 \times 0.14^2 + \left( \frac{3000}{10000} \right)^2 \times 0.02^2 + \left( \frac{4500}{10000} \right)^2 \times 0.08^2 \].Evaluating gives:\[ \sigma^2 = 0.00509 \].Thus, the standard deviation is:\[ \sigma = \sqrt{0.00509} \approx 0.0713 \].

Step 3: Calculate Variance With Covariance

For part (b), include the covariance between \(X_2\) and \(X_3\):\[ \sigma^2 = \left( \frac{2500}{10000} \right)^2 \times 0.14^2 + \left( \frac{3000}{10000} \right)^2 \times 0.02^2 + \left( \frac{4500}{10000} \right)^2 \times 0.08^2 + 2 \times \left( \frac{3000}{10000} \right) \times \left( \frac{4500}{10000} \right) \times (-0.005) \].Evaluating gives:\[ \sigma^2 = 0.00464 \].Thus, the standard deviation is:\[ \sigma = \sqrt{0.00464} \approx 0.0681 \].

Step 4: Compare and Comment

The mean rate of return remains the same in parts (a) and (b) at 0.084. The variance decreases in part (b) to 0.00464 from 0.00509 in part (a). This decrease illustrates a benefit of having a negative covariance, as it reduces the overall risk (as shown by the decreased variance).

Key Concepts

Mean Rate of Return

The mean rate of return is a fundamental concept in investment portfolio analysis. It represents the average expected return from your investment. Calculating the mean rate of return involves considering the weighted average of individual asset returns based on their allocation in the portfolio.
When you calculate the mean return for a portfolio, each asset's rate of return is multiplied by its proportion in the total investment. This results in a weighted average return. For example, if you invested $2500 in a stock with a return of 12%, $3000 in one with 4%, and $4500 in another with 7%, the mean rate of return for these assets is computed using their respective weights in the total $10,000 investment.
In our case, the mean rate of return for the entire portfolio was found to be 8.4%, representing how the combined investments are expected to perform over the year.

Variance Calculation

Variance is a measure used to quantify the risk associated with an investment. It reflects how much the returns of an asset differ from the mean return, indicating the degree of volatility.
To calculate the variance of a portfolio with independent returns, we look at the weighted sum of each asset's variance. First, we take each asset's variance, multiply it by its weight squared, and then sum them all up. This reflects independent risk contributions from each asset in the portfolio.
For our given assets, using their variances of 0.14, 0.02, and 0.08, we apply their respective weights to find the portfolio variance. For independent assets, we computed this value to be 0.00509. A higher variance indicates higher risk, while a lower variance points to more stable returns.

Covariance Impact

In portfolio analysis, covariance reveals the degree to which two assets move together. A crucial insight into managing risk. If assets' returns have negative covariance, they tend to move in opposite directions, thus stabilizing the portfolio's overall return.
When covariance is introduced, it impacts the variance calculation of the portfolio. Unlike independent returns where covariance does not factor in, a non-zero covariance can adjust your portfolio's overall risk. For example, a negative covariance between assets can reduce the variance, suggesting a reduction in risk due to their interactions.
In our scenario, incorporating a negative covariance of -0.005 between two assets led to a decrease in the portfolio variance from 0.00509 to 0.00464. This demonstrates how diversifying investments with negatively correlated assets can benefit a portfolio by reducing risk.

Normally Distributed Returns

Normally distributed returns describe a common statistical assumption in finance where asset returns are distributed in a symmetric, bell-shaped manner around the mean. This assumption simplifies risk management and return predictions since it relies on known statistical properties.
For normally distributed returns, the mean and variance are enough to depict the entire distribution. The normal distribution assumption allows investors to predict the likelihood of various return levels occurring.
In our problem setup, each asset's return was assumed to be normally distributed with specific means and variances. This helped in analyzing and calculating expected portfolio returns and risks effectively, knowing that certain percentiles of returns will lie within predictable bounds relative to the average.